2 Answers
2

Don't worry about the "$r$"'s. Let them hang along for the ride. Think of $r$ as just a specific number, if you like.

Let's start with the left hand side of your identity:
$$\tag{1}
(\color{maroon}2r\color{maroon}{\sin A\cos A})^2+r^2(\color{darkgreen}{\cos^2 A-\sin^2 A})^2
$$
and apply the double angle identities you've noted. Replacing, as we may, in $(1)$ the expression $\color{maroon}{2\sin A\cos A}$ with $\sin (2A)$ and the expression $\color{darkgreen}{\cos^2A-\sin^2A}$ with $\cos( 2A)$ we obtain
$$\tag{2}
\bigl(r\sin( 2A)\bigr)^2 +r^2\bigl(\cos(2A)\bigr)^2
$$

Using the rule $(ab)^2=a^2b^2$ and the notations $(\sin x)^2=\sin^2x$ and $(\cos x)^2=\cos^2 x$, we can write $(2)$ as
$$
\tag{3}
r^2\sin^2(2A)+r^2\cos^2(2A).
$$
Next, let's factor the $r^2$ term out. We can write $(3)$ as
$$\tag{4}
r^2\bigl(\sin^2 (2A)+\cos^2(2A)\bigr).
$$
But Pythagoras tell us that $\sin^2 (2A)+\cos^2 (2A)=1$; thus the expression in $(4)$ is simply $r^2$, which is what you wanted to show.

Double angle identities do shorten the calculation, but are unnecessary. Note that we are looking at
$$r^2\left(4\sin^2 A\cos^2 A +(\cos^2A-\sin^2 A)^2\right).\tag{$1$}$$
The following non-trigonometric identity is easy to verify, and quite useful:
$$(x-y)^2+4xy=(x+y)^2.\tag{$2$}$$
Let $x=\cos^2 A$ and $y=\sin^2 A$.
We get
$$4\sin^2 A\cos^2 A+(\cos^2 A-\sin^2 A)^2=(\cos^2 A+\sin^2 A)^2=1.$$